Vertex Operator Algebras Associated to Type G Affine Lie Algebras

TL;DR

This paper investigates the structure and representations of vertex operator algebras associated with affine G2 Lie algebras at specific admissible levels, establishing finiteness and reducibility results that support existing conjectures.

Contribution

It provides a detailed analysis of singular vectors, describes the associative algebra structure, and proves finiteness and complete reducibility of modules at certain levels.

Findings

Finite number of irreducible modules from category O.

Complete reducibility of modules at studied levels.

Supports existing conjecture on module structure.

Abstract

In this paper, we study representations of the vertex operator algebra $L(k,0)$ at one-third admissible levels $k= -5/3, -4/3, -2/3$ for the affine algebra of type $G_2^{(1)}$. We first determine singular vectors and then obtain a description of the associative algebra $A(L(k,0))$ using the singular vectors. We then prove that there are only finitely many irreducible $A(L(k,0))$-modules from the category $\mathcal O$. Applying the $A(V)$-theory, we prove that there are only finitely many irreducible weak $L(k,0)$-modules from the category $\mathcal O$ and that such an $L(k,0)$-module is completely reducible. Our result supports the conjecture made by Adamovi{\'c} and Milas in \cite{AM}.